sampling random signals in a fractional fourier tao zhang, wang, 2010)
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Sampling random signals in a fractional Fourier domain
Ran Tao n, Feng Zhang, Yue Wang
Department of Electronic Engineering, Beijing Institute of Technology, Beijing 100081, China
a r t i c l e i n f o
Article history:
Received 21 February 2010Received in revised form
12 November 2010
Accepted 20 November 2010
Keywords:
Random signals
Fractional Fourier transform
Fractional power spectrum
Fractional correlation
Multi-channel sampling
Periodic nonuniform sampling
a b s t r a c t
In this paper, we consider thesamplingand reconstruction schemesfor randomsignals in
the fractional Fourier domain. We define the bandlimited random signal in the fractionalFourier domain, and then propose the uniform sampling and multi-channel sampling
theorems forthe bandlimited randomsignal in thefractional Fourierdomain by analyzing
statistical properties of the input and the output signals for the fractional Fourier filters.
Our formulation and results are general and include derivative sampling and periodic
nonuniform samplingin thefractional Fourier domainfor randomsignalsas special cases.
& 2010 Elsevier B.V. All rights reserved.
1. Introduction
Digital signal processing relies on sampling a signal and
reconstructing it from its samples. Consequently, sampling
theory lies at the heart of signal processing as the digital
applications have developed rapidly over the last few
decades. Shannon sampling theory (also attributed to
Nyquist, Whittaker and Kotelnikov) is the milestone both
in terms of achievement and conciseness, which states that
for a complete reconstruction of an original bandlimited
signal the sampling rate must be at least twice the
maximum frequency present in the signal. (This is the
so-called Nyquist rate.) [1,2] The reconstruction formula
that complements the sampling theorem is
xðt Þ ¼X1
n ¼ À1 xðnT Þ sinpðt =T ÀnÞ
pðt =T ÀnÞ
where T is the sampling interval. In this widely used theorem,
the signal is assumed to be bandlimited or compact in the
Fourier domain. Thus, the sampling theorem associated with
the Fourier transform (FT) represents a signal in terms of
sinusoidals.
In fact, many natural signals are better represented in
alternative bases other than the Fourier basis. As the
fractional Fourier transform (FRFT) is a generalization of
the conventional Fourier transform and has found many
applications in optics and signal processing [3–11], the
study of the sampling theorems associated with the FRFT
has blossomed in recent years [12–18]. In [12], Xia firstly
shows that if a nonzero signal x(t ) is bandlimited in the
fractional Fourier domain with angle a, then it cannot be
bandlimited in the fractional Fourier domain with respect to
another angle b where ba7a+np for any integer n. Then,
the sampling expansion and spectral properties for a uni-
formly sampled signal bandlimited in the fractional Fourier
domain have been derived from different ways [12–15]. In
[16], the spectral analysis and reconstruction of a periodic
nonuniformly sampled signal bandlimited in the fractional
Fourier domain are presented. A more general sampling
theorem is considered in [17], where the multi-channel
sampling theorem in the fractional Fourier domain is also
studied. Recently, sampling and reconstruction of sparse
signals in thefractional Fourier domainhave been presented
[18]. The above sampling theorems assert that if a signal has
a narrower bandwidth or compact support in a fractional
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Signal Processing
0165-1684/$- see front matter & 2010 Elsevier B.V. All rights reserved.doi:10.1016/j.sigpro.2010.11.006
n Corresponding author.
E-mail addresses: [email protected] (R. Tao),
[email protected] (F. Zhang).
Signal Processing ] (]]]]) ]]]–]]]
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Fourierdomain,thenwe can usetheFRFTinstead of the FT to
sample a signal with a larger sampling interval. Therefore,
the classical sampling methods are not always efficient with
the possibility of unnecessary computational cost.
In practice, signals are often of random character. While
all previous generalized sampling approaches have typi-
cally dealt with the class of deterministic signals of known
finite spectral support in the fractional Fourier domain, wewill consider sampling methods for random signals with
known spectral densities. Firstly, we define a bandlimited
random signal in the fractional Fourier domain using the
fractional power spectrum. Then, we address the problem
of reconstructing the random signal with finite power
spectral support in the fractional Fourier domain from
uniform samples and multi-channel samples. Instead of
perfect reconstruction of the original signal in terms of the
samples, we determine the reconstruction through mini-
mization of the mean squared error (MSE) between the
signal and its reconstructed version. In particular, we
construct two kinds of multi-channel sampling structures
for fractional correlation functions. Our formulation andproof are general and include derivative sampling and
periodic nonuniform sampling in the fractional Fourier
domain for random signals as special cases.
2. Preliminaries
2.1. The fractional Fourier transform
The FRFT with angle a of a signal x(t ) is defined as [3,4]
X aðuÞ ¼ F a xðt Þ½ �ðuÞ ¼ exp À jpsgnðsinaÞ=4 Ã
exp ja=2Â Ã
ffiffiffiffiffiffi2pp
ffiffiffiffiffiffiffiffiffiffiffiffiffisina q ÂZ 1
À1e jðu2 þ t 2=2ÞcotaÀ jut csca xðt Þdt ð1Þ
where sgnðUÞ is the sign function. For a=0 and a=p/2, the
FRFT reduces to the identity transform and the conven-
tional FT, respectively. Note that the factor exp
À jpsgnðsinaÞ=4Â Ã
exp ja=2Â Ã
= ffiffiffiffiffiffi
2pp ffiffiffiffiffiffiffiffiffiffiffiffiffi
sina q
is sometimes
simplified as ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið1À j cotaÞ=2p
p .
The fundamental property of the FRFT is the angle
additivity property, which can be given by
F aþb xðt Þ½ �ðuÞ ¼ F a F b xðt Þ½ � ÃðuÞ ð2ÞBesides, the FRFT has the following important space
shift and phase shift properties [3]:
F a xðt ÀtÞ½ � ¼ X aðuÀtcosaÞe jðt2=2ÞsinacosaÀ jut sina ð3Þ
F a xðt Þe jvt h i
¼ X aðuÀv sinaÞeÀ jðv2=2Þsinacosaþ juv sina ð4Þ
where t and v represent the space and phase shift para-
meters, respectively. More details on the FRFT can be found
in [3,4].
2.2. The fractional power spectral density
For a random signal { x(t ),ÀNot oN}, its auto-correla-
tion function is defined by R xx(t 1,t 2)= R xx(t 2+t,t 2)= E [ x(t 1) xn
(t 2)]where E d½ � indicates the statistical expectation, t=t 1Àt 2, and
superscriptn is thecomplex conjugation.Motivated bythefact
that theFRFT generalizes theFT in a rotational manner, theath
fractional auto-correlation function of x(t ) is defined as [19]
Ra xxðtÞ ¼ limT -1
1
2T
Z T
ÀT
R xxðt 2 þt,t 2Þe jt 2tcota dt 2 ð5Þ
Then, the ath fractional power spectral density can be
given by [19]
P a xxðuÞ ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1þ j cota
2p
r F a Ra
xxðtÞÂ ÃðuÞeÀ jðu2=2Þcota ð6Þ
From(5) and (6) wecan see that botha timeaverage and
an ensemble average are in the fractional correlation func-
tion, and the fractional power spectrum is expressed in
terms of thefractional correlation function. Whena=p/2,(6)
becomes the Wiener–Khinchine theorem.
Filteringin a fractional Fourier domaincan be expressed
as [7,9]
s1ðt Þ ¼ F Àa F a s0ðt Þ½ � H aðuÞ
È Éwhere H a(u) is the transfer function, s0(t ) is the determi-nistic input signal and s1(t ) is the corresponding output
signal.
Let x(t ) and y(t ) be the random input and output of a
fractional Fourier filterH a(u), respectively; P a xxðuÞ and P a yyðuÞbe the fractional auto-power spectra of x(t ) and y(t ),
respectively; and P a yxðuÞ be the fractional cross-power
spectrum. Then the input–output relationships of the
fractional power spectral density for H a(u) are expressed
as [19]
P a yxðuÞ ¼ H aðuÞP a xxðuÞ ð7Þ
and
P a yyðuÞ ¼ H aðuÞ 2P a xxðuÞ ð8Þ
3. Sampling random signals in the fractional Fourier
domain
In this section, we treat the problem of reconstructing a
random signal that has compact support in the fractional
Fourier domainfrom a sequence of its uniform samples and
multi-channel samples. To analyze the sampling of random
signals in the fractional Fourier domain, we firstly give the
definition of a bandlimited random signal in the fractional
Fourier domain.
Definition 1. A randomsignal x(t ) is said tobe bandlimited
in the ath fractional Fourier domain if its fractional power
spectral density satisfies
P a xxðuÞ ¼ 0, 9u94ur ð9Þwhere ur is the smallest number such that (9) holds true
and is called the bandwidth of the random signal x(t )inthe
fractional Fourier domain.
3.1. Uniform sampling theorem for bandlimited random
signals in the fractional Fourier domain
Let a deterministic signal x(t ) be bandlimited in theath fractional Fourier domain with the bandwidth ur ,
Please cite this article as: R. Tao, et al., Sampling random signals in a fractional Fourier domain, Signal Process. (2010),doi:10.1016/j.sigpro.2010.11.006
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i.e., F a[ x(t )]=0 when 9u94ur . Then, the signal x(t ) can be
reconstructed from its uniform samples [12]:
xðt Þ ¼ eÀ jðt 2=2ÞcotaX1
n ¼ À1 xðnT Þe jðn2T 2=2Þcota sin ur cscaðt ÀnT Þ½ �
ur cscaðt ÀnT Þð10Þ
where the sampling interval T satisfies psina=T ¼ ur .It should be pointed that ur cscaðt ÀnT Þ ¼ pðt =T ÀnÞ.
Now, let us consider the case of uniform sampling for a
bandlimited random signal in the fractional Fourier
domain. We have the following sampling theorem:
Theorem 1. Let a random signal x(t) be bandlimited in the
ath fractional Fourier domain with the bandwidth ur . If its
chirped form, i.e., xðt Þe jðt 2=2Þcota, is stationary in the wide sense,
then x(t) can be reconstructed as
xðt Þ ¼ l:i:m: eÀ jðt 2=2ÞcotaX1
n ¼ À1 xðnT Þe jðn2 T 2=2Þcota
Âsin ur
ðt À
nT Þcsca
½ �ur ðt ÀnT Þcsca ð11Þwhere l:i:m: stands for limit in the mean square sense or
convergence in probability as well, i.e.,
limN -1
E xðt ÞÀeÀ jðt 2=2ÞcotaXN
n ¼ ÀN
xðnT Þe jðn2T 2=2Þcota
"(
 sin ur ðt ÀnT Þcsca½ �ur ðt ÀnT Þcsca
#2)
¼ 0
and sampling interval T ¼ ðpsina=ur Þ .
Proof. Let the estimate ^ xðt Þ be
^ xðt Þ ¼ eÀ jðt 2=2ÞcotaX1
n ¼ À1 xðnT Þe jðn2T 2=2Þcota sin ur ðt ÀnT Þcsca½ �
ur ðt ÀnT Þcsca
ð12Þ
Then, we have
E xðt ÞÀ^ xðt Þ à xÃðmT ÞÈ É¼ R xxðt ,mT ÞÀeÀ jðt 2=2Þcota
ÂX1
n ¼ À1R xxðnT ,mT Þe jðn2T 2=2Þcota sin ur ðt ÀnT Þcsca½ �
ur ðt ÀnT Þcscað13Þ
Since xðt Þe jðt 2=2Þcota is wide-sense stationary (WSS), we
have that
E xðt 1Þe jðt 21=2Þcota xÃðt 2ÞeÀ jðt 2
2=2Þcota
h i¼ e jððt 2 þtÞ2Àt 2
2=2ÞcotaE xðt 2 þtÞ xÃðt 2Þ½ �
¼ e jðt2=2Þcotaþ jt 2tcotaR xxðt 2 þt,t 2Þ ð14Þis only a function of the variable t where t=t 1Àt 2.
That is to say e jt 2tcotaR xxðt 2 þt,t 2Þ is only a function of the
variable t, then the fractional correlation function of x(t )
can be expressed as
Ra xxðtÞ ¼ limT -1
1
2T
Z T
ÀT
R xxðt 2 þt,t 2Þe jt 2tcota dt 2
¼R xx
ðr
þt,r
Þe jrtcota
ð15
Þwhere the equation is valid for all r.
Substituting (15) into (13), we can deduce that
E xðt ÞÀ^ xðt Þ à xÃðmT ÞÈ É¼ Ra xxðt ÀmT ÞeÀ jmT ðt ÀmT ÞcotaÀeÀ jðt2=2Þcota
ÂX1
n ¼ À1Ra xxðnT ÀmT Þe jðn2T 2=2ÞcotaÀ jmT ðnT ÀmT Þcota
(
Â
sin ur ðt ÀnT Þcsca½ �
ur ðt ÀnT Þcsca):
ð16
ÞAccording to (3), (4) and (6), the FRFT of the fractional
correlation function with space shift and phase shift can be
written as
F a Ra xxðtÀt0ÞeÀ jt0tcotah i
¼ F a Ra xxðtÞÂ Ã
eÀ jðt20=2ÞcotaÀ jut0 csca
¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2p
1þ j cota
s P a xxðuÞe jðu2=2ÞcotaÀ jðt2
0=2ÞcotaÀ jut0 csca
Note that the random signal x(t ) is bandlimited in the ath
fractional Fourier domain, i.e., the ath fractional power
spectral density of the signal x(t ) obeys (9). Thus theshifted
fractional correlation function Ra xxðtÀt0ÞeÀ jt0tcota is ban-dlimited in the ath fractional Fourier domain with the
bandwidth ur . Applyingthe uniform sampling expansionto
the deterministic function Ra xxðtÀt0ÞeÀ jt0tcota yields
Ra xxðtÀt0ÞeÀ jt0tcota ¼ eÀ jðt2=2ÞcotaX1
n ¼ À1e jðn2T 2=2ÞcotaRa xxðnT Àt0Þ
ÂeÀ jt0 nT cota sin ur ðtÀnT Þcsca½ �ur ðtÀnT Þcsca
ð17Þ
Interchanging the variables t=t and t0=mT in (17), we
obtain
Ra xx
ðt À
mT ÞeÀ jmTt cota
¼eÀ jðt 2=2Þcota X
1
n ¼ À1e jðn2 T 2=2ÞcotaRa
xx
ðnT
ÀmT
ÞÂeÀ jmnT 2 cota sin ur ðt ÀnT Þcsca½ �
ur ðt ÀnT Þcscað18Þ
Substituting (18) into (16), we deduce
E xðt ÞÀ^ xðt Þ à xÃðmT ÞÈ É¼ 0 ð19Þ
This means that, for every m, xðt ÞÀ^ xðt ÞÂ Ãis orthogonal to
x(mT ). Since ^ xðt Þ is a linear summation of x(mT ), xðt ÞÀ^ xðt ÞÂ Ãis also orthogonal to ^ xðt Þ, i.e.,
E xðt ÞÀ^ xðt Þ Ã^ xÃðt ÞÈ É¼ 0 ð20Þ
On the other hand,
E xðt ÞÀ^ xðt Þ à xÃðt ÞÈ É¼ R xxðt ,t ÞÀeÀ jðt2=2Þcota
X1n ¼ À1
R xxðnT ,t Þ
Âe jðnT 2=2Þcota sin ur ðt ÀnT Þcsca½ �ur ðt ÀnT Þcsca
¼ Ra xxð0ÞÀX1
n ¼ À1Ra xxðnT Àt Þe jððnT Àt Þ2=2Þcota
 sin ur ðt ÀnT Þcsca½ �ur ðt ÀnT Þcsca
ð21Þ
Similarly, choosing the variables t=t0=t in (17), we
obtain
Ra xxð0ÞeÀ jt
2cota ¼ eÀ jðt
2
=2Þcot
aX1
n ¼ À1 e jðn
2
T
2
=2Þcot
aRa xxðnT Àt Þ
Please cite this article as: R. Tao, et al., Sampling random signals in a fractional Fourier domain, Signal Process. (2010),doi:10.1016/j.sigpro.2010.11.006
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ÂeÀ jtnT cota sin ur ðt ÀnT Þcsca½ �ur ðt ÀnT Þcsca
ð22Þ
Substituting (22) into (21) yields
E xðt ÞÀ^ xðt Þ à xÃðt ÞÈ É¼ 0 ð23Þ
Therefore, combining (20) and (23), we can obtain
E xðt ÞÀ^ xðt Þ 2h i¼ E xðt ÞÀ^ xðt ÞÂ ÃxÃðt ÞÀ^ xÃðt ÞÂ ÃÈ É
¼ E xðt ÞÀ^ xðt ÞÂ Ã xÃðt ÞÈ ÉÀE xðt ÞÀ^ xðt ÞÂ Ã
^ xÃðt ÞÈ É¼ 0:
This concludes the proof of the theorem. ’
Remarks. Theorem 1 can be developed and understood
from another aspect.
Let xc (t ) denote the chirped form of the signal x(t ), i.e.,
xc ðt Þ ¼ xðt Þe jðt 2=2Þcota ð24ÞSince the random signal x(t ) is bandlimited in the ath
fractional Fourier domain with the bandwidth ur ,by(6)and
(9) we can derive
Ra xxðtÞ ¼Z ur
Àur
P a xxðuÞe jðu2=2Þcotah i
eÀ jðu2 þt2Þ=2 cotaþ jutcscadu
¼Z ur
Àur
P a xxðuÞeÀ jðt2=2Þcotaþ jutcscadu ð25Þ
Since xc (t ) is WSS, we can obtain its auto-correlation
function from (14) and (15):
R xc xc ðtÞ ¼ e jt2
2cotaþ jt 2tcotaR xxðt 2 þt,t 2Þ ¼ e jt
2
2cotaRa xxðtÞ ð26Þ
Combining (25) and (26), we have
R xc xc ðtÞ ¼ Ra xxðtÞe jðt2=2Þcota ¼Z
ur
Àur
P a xxðuÞe jutcscadu ð27Þ
Wecan see that xc (t ) is conventionally bandlimited with the
bandwidth ur csca. Therefore, applying the classical recon-
structionto the bandlimited random signal xc (t ), wecan obtain
(11). In fact, we can utilize the chirp multiplication (Q) and
FRFT (F a) operators for the development [4].
Eq. (10)establishes the relationship between an original
random signal and its uniform samples. From Theorem 1,
for a bandlimited random signal in the fractional Fourier
domain, we can reconstruct the original signal in terms of
its uniform samples x(nT ) in mean square sense, provided
the sampling interval satisfies T rðpsina=ur Þ.
3.2. Multi-channel sampling theorem for bandlimited
random signals in the fractional Fourier domain
There is a variety of applications in which a signal is
sampled in other ways, such as derivative sampling or
periodic nonuniform sampling[20,21]. In [22], Papoulis has
introduced the multi-channel sampling (generalized sam-
pling) theorem, and the derivative sampling and the
periodic nonuniform sampling are typical instances of it.
In [17], the generalized sampling theorem for deterministic
signals bandlimited in the fractional Fourier domain is
obtained. Multi-channel sampling theorem in the frac-
tional Fourier domain is a generalized sampling theorem.Many sampling strategies in the fractional Fourier domain
can be regarded as special cases of this generalized
sampling method such as derivative sampling and periodic
nonuniform sampling in the fractional Fourier domain. In
this part, we consider this generalized sampling for the
bandlimited random signal in the fractional Fourier
domain. We have the following theorem:
Theorem 2. Let a random signal x(t) be bandlimited in theath fractional Fourier domain with the bandwidth ur , and its
chirped form, i.e., xðt Þe jðt 2=2Þcota, be stationary in the wide
sense.If therandom signalx(t) is processedby M ath fractional
Fourier filters H a,k(u) resulting M outputs g k(t), k ¼ 1,. . .,M ,
then x(t) canbe reconstructed in terms of the samples g k(nT) in
the mean square sense:
xðt Þ ¼ l:i:m:eÀ jðt 2=2ÞcotaX1
n ¼ À1
XM
k ¼ 1
g kðnT Þe jðn2T 2=2Þcota ykðt ÀnT Þ
ð28Þwhere l:i:m: stands for limit in the mean square sense or
convergence in probability as well, i.e.,
limN -1
E xðt ÞÀeÀ jðt 2=2ÞcotaXN
n ¼ ÀN
XM
k ¼ 1
g kðnT Þe jðn2T 2=2Þcota ykðt ÀnT Þ
" #2
8<:
9=;¼ 0
and sampling interval is T ¼ ðM psina=ur Þ .
Here, reconstruction kernel functions ykðt Þ ¼ ð1=c Þ R Àur þ c Àur
Y kðu,t Þe jut cscadu, k ¼ 1,. . .,M , and M functions Y k(u,t) are
obtained by solving the following M linear equations:
H 1ðuÞY 1ðu,t Þ þ Á Á Á þH M ðuÞY M ðu,t Þ ¼ 1
H 1ðuþc ÞY 1ðu,t Þ þ Á Á Á þ H M ðuþc ÞY M ðu,t Þ ¼ e jct csca
Á Á Á Á Á Á Á Á Á Á Á Á Á Á Á Á Á Á Á Á Á Á Á Á Á Á Á Á Á ÁH 1
½u
þðM
À1Þc �Y 1
ðu,t
Þ þ Á Á Á þH M
½u
þðM
À1Þc �Y M
ðu,t
Þ ¼e jðM À1Þct csca
8>>>><
>>>>:ð29Þ
where Àur rurÀur +c and c ¼ ð2ur =M Þ.
Proof. Rewriting the fractional Fourier filter in the form
H a,kðuÞ ¼ H !
a,kðuÞeÀ jðu2=2Þcota, we have
Ga,kðuÞ ¼ X aðuÞH a,kðuÞ ¼ X aðuÞ H !
a,kðuÞeÀ jðu2=2Þcota ð30Þ
where Ga,k(u) is the ath FRFT of g k(t ).
From the definition of the FRFT, we have
Ga,kðuÞ ¼ eÀ jðu2=2Þcota
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1À jcota
2pr Z 1
À1
e jðu2 þ t 2=2ÞcotaÀ jut csca xðt Þdt
 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1À jcota
2p
r Z 1
À1e jðu2 þv2=2Þ cotaÀ juvcsca h
!kðvÞdv
¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1À jcota
2p
r 2 Z 1
À1
Z 1
À1e jðu2 þ t 2 þv2=2ÞcotaÀ juðt þ vÞcsca
 xðt Þ h!
kðvÞdtdv
By making the change of variable v = z Àt , wecanobtain a
convolution form of (30):
g kðt Þe jðt 2=2Þcota ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1À j cota
2p
r xðt Þe jðt 2=2Þcotaà h
!kðt Þe jðt 2=2Þcota
ð31
Þwhere n denotes the conventional convolution operator.
Please cite this article as: R. Tao, et al., Sampling random signals in a fractional Fourier domain, Signal Process. (2010),doi:10.1016/j.sigpro.2010.11.006
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Note that xðt Þe jðt 2=2Þcota is a wide-sense stationary random
signal. Then, from (31) we canobtain that g kðt Þe jðt 2=2Þcota is also
wide-sense stationary, and random signals xðt Þe jðt 2=2Þcota and
g kðt Þe jðt 2=2Þcota are jointly wide-sense stationary.
From (6) and (7), we have
P a g k , xðuÞ ¼ H a,kðuÞP a xxðuÞ ð32Þ
and
F a Ra g k , xðtÞh i
¼ H a,kðuÞF a Ra x, xðtÞÂ Ã
: ð33Þ
From (33) we can see that, the deterministic fractional
correlation function Ra x, xðtÞ can beseen astheinputof the M
fractional Fourier filters H a,k(u), k ¼ 1,. . .,M , and conse-
quently results in M deterministic outputs Ra g k , xðtÞ. We
show this important relation in Fig. 1, where T a,kðuÞ ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið1À j cota=2pÞ
p R 1À1 ykðt ÞeÀ jut cscadt .
Therefore, applying the generalized sampling theorem in
the fractional Fourier domain to the fractional correlationfunction Ra x, xðtÞ, we can obtain
Ra x, xðtÞ ¼ eÀ jðt2=2ÞcotaX1
n ¼ À1
XM
k ¼ 1
Ra g k , xðnT Þe jðn2T 2=2Þcota ykðtÀnT Þ
ð34Þ
From (31), we can obtain its time-delay version:
g kðt Àt0Þe jððt Àt0Þ2=2Þcota
¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1À j cota
2p
r xðt Àt0Þe jððt Àt0Þ2=2Þcotaà h
!kðt Þe jðt 2=2Þcota ð35Þ
which can be further written as
g kðt Àt0Þe jððt20À2t t0Þ=2Þcota
h ie jðt 2=2Þcota
¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1À jcota
2p
r xðt Àt0Þe jðt2
0À2t t0Þ=2Þcota
h iÂe jðt 2=2Þcotaà h
!kðt Þe jðt 2=2Þcota ð36Þ
According to (36), we can rewrite (34) as
Ra x, xðtÀt0Þe jððt20À2tt0Þ=2Þcota ¼ eÀ jðt2=2Þcota
X1n ¼ À1
XM
k ¼ 1
Ra g k , xðnT Àt0Þ
Âe jððt20À2nT t0Þ=2Þcotae jðn2T 2=2Þcota ykðtÀnT Þ ð37Þ
Let the estimate
^ xðt Þ ¼ eÀ jðt 2=2ÞcotaX1
n ¼ À1
XM
k ¼ 1
g kðnT Þe jðn2T 2=2Þcota ykðt ÀnT Þ
ð38ÞAccording to the joint stationarity of the random signals
xðt Þe j
ðt 2=2
Þcota and g
kðt Þe j
ðt 2=2
Þcota, we can derive
E xðt ÞÀ^ xðt Þ à xÃðt ÞÈ É¼ R x, xðt ,t ÞÀeÀ jðt 2=2Þcota
ÂX1
n ¼ À1
XM
k ¼ 1
R g k , xðnT ,t Þe jððnT Þ2=2Þcota ykðt ÀnT Þ
¼ Ra x, xð0ÞÀeÀ jðt 2=2Þcota
ÂX1
n ¼ À1
XM
k ¼ 1
R g k , xðnT ,t Þe jððnT Þ2=2Þcota ykðt ÀnT Þ
¼ Ra x, xð0ÞÀeÀ jðt 2=2Þcota
ÂX1
n ¼ À1
XM
k ¼ 1
Ra g k , xðnT Àt ÞeÀ jt ðnT Àt Þe jððnT Þ2=2Þcota ykðt ÀnT Þ:
ð39
ÞLet t=t0=t in (37) and then substitute the result into (39):
E xðt ÞÀ^ xðt Þ à xÃðt ÞÈ É¼ 0 ð40Þ
On the other hand, according to (38), for the samples
g l(mT ) of the lth output we have
E xðt ÞÀ^ xðt ÞÂ Ã g l
ÃðmT ÞÈ É¼ R x, g l ðt ,mT ÞÀeÀ jðt 2=2Þcota
X1n ¼ À1
XM
k ¼ 1
R g k , g l ðnT ,mT Þ
Âe jððnT Þ2=2Þcota ykðt ÀnT Þ ð41ÞAccording to the properties of the fractional correlation
function Ra x, g lðtÞ, we have
F a Ra g k , g lðtÞ
h i¼ H a,kðuÞF a Ra x, g l
ðtÞh i
, k ¼ 1,. . .,M ð42Þ
and
P a g k , g lðuÞ ¼ H a,kðuÞP a x, g l
ðuÞ, k ¼ 1,. . .,M ð43Þ
From (43) we can see that the deterministic fractional
correlation function Ra x, g lðtÞ can beseen asthe input ofthe M
fractional Fourier filters H a,k(u), k ¼ 1,. . .,M , and results in
M deterministic outputs Ra g k , g l
ðtÞ. We show this important
relation in Fig. 2.
Fig. 1. The generalized sampling configuration where the input is Ra x, xðtÞ. Fig. 2. The generalized sampling configuration where the input is Ra x, g lðtÞ.
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Therefore, applying the generalized sampling theorem in
the fractional Fourier domain to the fractional correlation
function Ra x, g lðtÞ, we can obtain
Ra x, g lðtÞ ¼ eÀ jðt2=2Þcota
X1n ¼ À1
XM
k ¼ 1
Ra g k , g lðnT Þe jðn2 T 2=2Þcota ykðtÀnT Þ
ð44
ÞSimilar to (37), we can further obtain
Ra x, g lðtÀt0Þe jððt2
0À2tt0Þ=2Þcota ¼ eÀ jðt2=2Þcota
ÂX1
n ¼ À1
XM
k ¼ 1
Ra g k , g lðnT Àt0Þe jððt2
0À2nT t0Þ=2Þcotae jðn2T 2=2Þcota ykðtÀnT Þ
ð45ÞReplacing t=t and t0=mT in (45), and then substituting
the result into (41) yields
E xðt ÞÀ^ xðt ÞÂ Ã
g lÃðmT Þ
È É¼ R x, g l ðt ,mT ÞÀeÀ jðt 2=2Þcota
ÂX1
n ¼ À1
XM
k ¼ 1
R g k , g l ðnT ,mT Þe jððnT Þ2=2Þcota ykðt ÀnT Þ
¼ Ra x, g lðt ÀmT ÞeÀ jmT ðt ÀmT ÞcotaÀeÀ jðt 2=2Þcot a
ÂX1
n ¼ À1
XM
k ¼ 1
Ra g k , g lðnT ÀmT ÞeÀ jmT ðnT ÀmT Þcota
Âe jððnT Þ2=2Þcota ykðt ÀnT Þ ¼ 0 ð46ÞFrom (46) we can obtain that for every m and l, 1r lrM ,
xðt ÞÀ^ xðt ÞÂ Ãand g l(mT ) are orthogonal. Since ^ xðt Þ is a linear
summation of the signals g l(mT ), we have
E xðt ÞÀ^ xðt Þ Ã^ xÃðt ÞÈ É¼ 0 ð47Þ
Combining (40) and (47), we can obtain
E xðt ÞÀ^ xðt Þ 2h i
¼ E xðt ÞÀ^ xðt ÞÂ ÃxÃðt ÞÀ^ x
Ãðt Þ ÃÈ É¼ E xðt ÞÀ^ xðt Þ Ã
xÃðt ÞÈ ÉÀE xðt ÞÀ^ xðt Þ Ã^ xÃðt ÞÈ É¼ 0
This completes the proof. ’
From Theorem 2, for a bandlimited randomsignal in the
fractional Fourier domain, we can reconstruct the original
signal in terms of generalized samples g l(nT ), 1r lrM in
the sense of mean square.
Note that Theorem 2 has a similar form with the
classical result in Papoulis’ work [Eq. (39), 22].Since derivative sampling and periodic nonuniform
sampling strategies in the fractional Fourier domain can
be seen as the special cases of the generalized sampling
method in the fractional Fourier domain by choosing
special fractional Fourier filters H a,k(u), we can obtain
the reconstruction methods for bandlimited random sig-
nals in the fractional Fourier domain from derivative
samples and periodic nonuniform samples that are listed
as follows.
Corollary 1. Let a random signal x(t) be bandlimited in the
ath fractional Fourier domain with the bandwidth u r . If its
chirped form, i.e., xðt Þe jðt 2=2
Þcota
, is stationary in the wide sense,then x(t) can be reconstructed from the samples of the signal
and its derivatives:
xðt Þ ¼ l:i:m: eÀ jðt 2=2ÞcotaX1
n ¼ À14sin
2 saðt ÀnT Þcsca
2
!
Âe jððnT Þ2=2Þcota xðnT Þðsaðt ÀnT ÞcscaÞ2
þ xuðnT Þsinaþ j nTxðnT Þcosa
s2aðt ÀnT Þcsca
" #
ð48Þ
where the sampling interval is T ¼ ð2psina=ur Þ.
Proof. Since the FRFT has the following properties [3]:
F a xuðt Þ½ � ¼ X uaðuÞcosaþ juX aðuÞsina
F a txðt Þ½ � ¼ uX aðuÞcosaþ jX uaðuÞsina
then
F a xuðt Þsinaþ j txðt Þcosa½ � ¼ juX aðuÞHence, by considering the case of Theorem 2 for M =2
with H a,1(u)= 1 and H a,2(u)= ju, and after some derivations,
we can obtain the result. ’
Corollary 2. Consider a periodic nonuniform sampling scheme: the sampling points are divided into groups of M
points with each group having a period of T. Denoting the
points in one period by t k, k ¼ 1,. . .,M , the complete set of
sample points can be written as nT+t k, n ¼ . . .,À1,0,1,. . .,
k ¼ 1,. . .,M , where T ¼ M psina=ur . Let a random signal x(t)
be bandlimited in the ath fractional Fourier domain with the
bandwidth ur . If its chirped form, i.e., xðt Þe jðt 2=2Þcota, is
stationary in the wide sense, then x(t) can be reconstructed
from its periodic nonuniform samples:
xðt Þ ¼ l:i:m: eÀ jðt 2=2ÞcotaXM
k ¼ 1
X1n ¼ À1
xðnT þt kÞe jððnT þ t kÞ2=2Þcota
 ðÀ1ÞnM QM
q ¼ 1 sin½pðt Àt qÞ=T �ðpðt ÀnT Àt kÞ=T ÞQM
q ¼ 1,qak sin½pðt kÀt qÞ=T �ð49Þ
where nT+t k are sampling points and T ¼ ðM psina=ur Þ.Proof. According to (3) and (4), we have
F a xðt þt kÞe jt k cota t h i
¼ X aðuÞeÀ jðt 2k=2Þcotaþ jut k csca
Then, by choosing H a,kðuÞ ¼ eÀ jðt 2k=2Þcotaþ jut k csca in the case
of Theorem 2 we can obtain the result. ’
4. Conclusion
In this paper, we have treated the problem of samplingand reconstruction of random signals in the fractional
Fourier domain. We have shown that for bandlimited
random signals in the fractional Fourier domain, the
original signal can be reconstructed from its uniform
samples and multi-channel samples in MSE sense. Our
formulation and proof are general, and include derivative
sampling and periodic nonuniform sampling in the frac-
tional Fourier domain for random signals as special cases.
Acknowledgements
This work was supported in part by the National ScienceFoundation of China for Distinguished Young Scholars under
Please cite this article as: R. Tao, et al., Sampling random signals in a fractional Fourier domain, Signal Process. (2010),doi:10.1016/j.sigpro.2010.11.006
R. Tao et al. / Signal Processing ] (]]]]) ]]]–]]]6
8/6/2019 Sampling Random Signals in a Fractional Fourier tao Zhang, Wang, 2010)
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Grant 60625104, the National Natural Science Foundation of
China under Grants 60890072 and60572094 andtheNational
Key Basic Research Program Founded by MOST under Grant
2009CB724003.
References
[1] A.J. Jerri,The Shannon sampling theorem—
its various extensionsandapplications: a tutorial review, Proc. IEEE 65 (11) (1977)1565–1596.
[2] M. Unser, Sampling-50 years after Shannon, Proc. IEEE 88 (4) (2000)569–587.
[3] L.B. Almeida, The fractional Fourier transform and time–frequencyrepresentations, IEEE Trans. Signal Process 42 (11) (1994)3084–3091.
[4] H.M. Ozaktas, Z. Zalevsky, M.A. Kutay, The Fractional Fourier Trans-form with Applications in Optics and Signal Processing, Wiley,New York, 2000.
[5] D. Mendlovic, H.M. Ozaktas, Fractional Fourier transforms and theiroptical implementation: I, J. Opt. Soc. Am. A 10 (1993) 1875–1881.
[6] H.M. Ozaktas, D. Mendlovic, Fractional Fourier transforms and theiroptical implementation: II, J. Opt. Soc. Am. A 10 (1993) 2522–2531.
[7] H.M. Ozaktas, B. Barshan, D. Mendlovic, L. Onural, Convolution, filtering,andmultiplexing in fractional Fourier domains andtheir relation to chirpand wavelet transforms, J. Opt. Soc. Am. A 11 (1994) 547–559.
[8] H.M. Ozaktas, O. Arikan, M.A. Kutay, G. Bozdagi, Digital computationof the fractional Fourier transform, IEEE Trans. Signal Process. 44 (9)(1996) 2141–2150.
[9] M.A. Kutay, H.M. Ozaktas, O. Arikan, L. Onural, Optimal filtering infractional Fourier domains, IEEE Trans. Signal Process. 45 (5) (1997)1129–1143.
[10] I.S. Yetik, A. Nehorai, Beamforming using the fractional Fouriertransform, IEEE Trans. Signal Process. 51 (6) (2003) 1663–1668.
[11] H.-B. Sun, G.-S. Liu, H. Gu, W.-M. Su, Application of the fractional
Fourier transform to moving target detection in airborne SAR, IEEETrans. Aerosp. Electron. Syst. 38 (4) (2002) 1416–1424.
[12] X.-G. Xia, On bandlimited signals with fractional Fourier transform,IEEE Signal Process. Lett. 3 (3) (1996) 72–74.
[13] R. Torres, P. Pellat-Finet, Y. Torres, Sampling theorem for fractionalbandlimited signals: a self-contained proof. Application todigital holography, IEEE Signal Process. Lett. 13 (11) (2006)676–679.
[14] R. Tao, B. Deng, W.-Q. Zhang, Y. Wang, Sampling and sampling rateconversion of band-limited signals in the fractional Fourier trans-form domain, IEEE Trans. Signal Process 56 (1) (2008) 158–171.
[15] C. Candan, H.M. Ozaktas, Sampling and series expansion theoremsfor fractional Fourier and other transforms, Signal Process. 83 (11)(2003) 2455–2457.
[16] R. Tao, B.-Z. Li, Y. Wang, Spectral analysis and reconstruction for
periodic nonuniformly samples signals in fractional Fourier domain,IEEE Trans. Signal Process. 55 (7) (2007) 3541–3547.
[17] F. Zhang, R. Tao, Y. Wang, Multi-channel sampling theorems forband-limited signals with fractional Fourier transform, Sci. China 51(6) (2008) 790–802.
[18] A. Bhandari, P. Marziliano, Sampling and reconstruction of sparsesignals in fractional Fourier domain, IEEE Signal Process. Lett. 17 (3)
(2010) 221–224.[19] R. Tao, F. Zhang, Y. Wang, Fractional power spectrum, IEEE Trans.
Signal Process. 56 (9) (2008) 4199–4206.
[20] Y.C. Eldar, A.V. Oppenheim, Filterbank reconstruction of bandlimitedsignals from nonuniform and generalized samples, IEEE Trans.Signal Process 48 (10) (2000) 2864–2875.
[21] Y.-C. Jenq, Digital spectra of nonuniformly sampled signals: funda-mentals and high-speed waveform digitizers, IEEE Trans. Instrum.Meas. 37 (2) (1988) 245–251.
[22] A. Papoulis, Generalized sampling expansion, IEEE Trans. Circ. Syst.
CAS-24 (11) (1977) 652–654.
Please cite this article as: R. Tao, et al., Sampling random signals in a fractional Fourier domain, Signal Process. (2010),doi:10.1016/j.sigpro.2010.11.006
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